Drawing and computing external rays in the multiple-spiral medallions of the Mandelbrot set

Abstract

The multiple-spiral medallions are beautiful decorations of the Mandelbrot set. Computer graphics provide an invaluable tool to study the structure of these decorations with point symmetry (i.e., the medallions can be rotated 180° about a central point to form an identical structure). The multiple-spiral medallions are formed by an infinity of baby Mandelbrot sets. Until now, the external arguments of the external rays landing at the cusps of the cardioids of these baby Mandelbrot sets could not be calculated. Recently, a new algorithm has been proposed in order to calculate the external arguments in the Mandelbrot set. In this paper, we use an extension of this algorithm for the calculation of the binary expansions of the external arguments of the baby Mandelbrot sets in the multiple-spiral medallions. © 2008 Elsevier Ltd. All rights reserved.